Is negative-weight percolation compatible with SLE?

TL;DR

This study investigates the geometric properties of paths in the negative-weight percolation model on 2D lattices, finding they do not conform to Schramm-Loewner evolution predictions.

Contribution

It provides a numerical analysis of the paths' geometry in NWP, testing their compatibility with SLE, which was previously unexplored.

Findings

Paths do not follow SLE predictions based on fractal dimension.

Schramm's left passage formula does not hold for these paths.

The geometric properties suggest a different universality class.

Abstract

We study numerically the geometrical properties of minimally weighted paths that appear in the negative-weight percolation (NWP) model on two-dimensional lattices assuming a combination of periodic and free boundary conditions (BCs). Each realization of the disorder consists of a random fraction 1-rho of bonds with unit strength and a fraction rho of bond strengths drawn from a Gaussian distribution with zero mean and unit width. For each such sample, the path is forced to span the lattice along the direction with the free BCs. The path and a set of negatively weighted loops form a ground state (GS). A ground state on such a lattice can be determined performing a non-trivial transformation of the original graph and applying sophisticated matching algorithms. Here we examine whether the geometrical properties of the paths are in accordance with predictions of Schramm-Loewner evolution (SLE). Measuring the fractal dimension and reviewing Schramm's left passage formula indicates that the paths cannot be described in terms of SLE.